Abstract: Sums of squares have been a research topic for as long as people have studied algebra and number theory. In modern language, some of the central questions are as follows. Let R be a ring with 1. Which elements in R can be written as sums of squares of elements in R? If an element is a sum of squares, how many squares are needed to write it as such? We give a survey of a few (of the many) classical and more recent results and open problems, focusing on fields, simple (or division) algebras and commutative rings.

Abstract: In 1911 Hermann Weyl proved his famous law on the asymptotic distribution of eigenvalues of Laplacians on a bounded domain. Answering a question posed by physicists, Lorentz and Sommerfeld among others, he showed that the statistics of large eigenvalues determines the  volume and dimension of such a domain. This result, which nowadays is paraphrased as``one can hear the volume and dimension of a bounded domain", is the foundation stone of a remarkable edifice  of modern  mathematics known as spectral geometry. The  ultimate goal is to know how much of the geometry and topology of a Riemannian manifold is encoded in its spectrum. Forexample, is it true that isospectral manifolds are isometric? That is,  Can one hear the shape of a drum? While the answer is in general negative, we  know that one can  hear the total scalar curvature, Betti numbers, and signature   of a closed Riemannian manifold. In fact an  infinite sequence of spectral invariants, known as DeWitt--Gilkey--Seeley coefficients can be defined and  computed from the short time asymptotic of the heat kernel. Another set of ideas in spectral geometry concerns with different types of trace formulae and applications to number theory, quantum physics, and quantum chaos. In some sense this even goes back to the very origins of the Weyl  law in quantum mechanics and in deriving Planck's radiation formula from it.

In this talk I shall outline some of these connections and then focus on our current joint work with Farzad Fathizadeh and show how some of these ideas can be imported to the world of noncommutative geometry of Alain Connes. In fact without spectral geometry there could be no noncommutative geometry!  In particular I shall highlight our recent proof of a Weyl law for noncommutative tori equipped with a general metric.